Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13122 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C2×D4 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C8×D4 |
| prime radical | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ C22×C16×C19 ◁ G | |
326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,22)(24,25,30,35,31,34,49,52,28,33,37,38,29,46,48,54)(26,36,41,53,42,47,51,55,39,27,43,45,40,44,32,50);(20,21,23,22)(24,26,37,43,31,42,48,32,28,39,30,41,29,40,49,51)(25,27,38,53,34,44,54,55,33,36,35,45,46,47,52,50);(20,22,23,21)(24,27,28,36)(25,32,33,51)(26,38,39,35)(29,47,31,44)(30,45,37,53)(34,41,46,43)(40,52,42,54)(48,55,49,50)1444 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6110);